构造辅助函数法在微积分证明中的教学研究

Research on the Application of the Construction of Auxiliary Function Method in the Calculus Proof

在微积分知识学习时,通常在证明某个问题的结论时,通过已有的条件无法直接推导所证的结论,这时可以尝试运用构造函数法,根据命题中的条件,将结论变换,从而构造出一个辅助函数,再运用有关的定理结论推导出命题的结论,从而能对命题的证明起到事半功倍的效果.构造函数法是一种重要的数学方法,其构造方法思路也是多种多样的,文章通过构造函数法在一些著名的定理,公式以及经典例题的运用,尝试找出如何构造辅助函数的几种方法,并通过这些方法在一些具体实例中的运用,归纳出构造函数法的一些思路.

In the study of advanced mathematics calculus knowledge,when we tend to prove the conclusion of a problem,we will meet difficulties which mean that we can＇t come to the direct conclusion through existing conditions.In the case of this situation,we can usually apply to the structure method of auxiliary.According to the conditions in the proposition,we can transform the conclusion,thus constructing an auxiliary function and then use related theorem to prove the conclusion.In this way,we can achieve a multiplier effect on the proof.The constructor method is a kind of important mathematical method.Its constructive methods and ideas are various.This paper tries to find out some ways of how to construct auxiliary function through its application to some famous theorem、formula and typical examples,and then we can conclude some new ideas of auxiliary function by applying these ways to some practical examples.